Invertible positive maps that are not automorphism

TL;DR

This paper presents a method to construct invertible linear maps that preserve a cone but whose inverses do not, illustrating that positive invertible maps need not be automorphisms.

Contribution

It introduces a novel construction of positive invertible maps that are not automorphisms, with explicit examples and a perturbation approach.

Findings

Constructed invertible maps with cone-preserving but inverse non-preserving properties

Showed existence of rank-one perturbations of automorphisms that are positive but not invertible

Provided diverse examples illustrating these phenomena

Abstract

Let $X$ be a real normed vector space with a cone $K\subseteq X$ satisfying either (i) $K$ is closed with non-empty interior or (ii) $K$ has non-zero extremals or (iii) $K$ is closed and $X$ is a Banach space. In this short note, we provide a method to construct an invertible linear map $T\colon X\to X$ such that $T[K]\subseteq K$ but $T^{-1}[K]\not\subseteq~K$. In particular, we show that, for every cone automorphism $S\colon X\to X$, there exists a rank one perturbation of $S$ which is positive and invertible, but does not have a positive inverse. We provide examples from four diverse situations.